# Status of local gauge invariance in axiomatic quantum field theory

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In his recent review...

• Sergio Doplicher, The principle of locality: Effectiveness, fate, and challenges, J. Math. Phys. 51, 015218 (2010), doi

...Sergio Doplicher mentions an important open problem in the program of axiomatic quantum field theory, quotes:

In the physical Minkowski space, the study of possible extensions of superselection theory and statistics to theories with massless particles such as QED is still a fundamental open problem.

...

More generally the algebraic meaning of quantum gauge theories in terms local observables is missing. This is disappointing since the success of the standard model showed the key role of the gauge principle in the description of the physical world; and because the validity of the principle of locality itself might be thought to have a dynamical origin in the local nature of the fundamental interactions, which is dictated by the gauge principle combined with the principle of minimal coupling.

While it is usually hard enough to understand the definite results of a research program, it is even harder if not impossible, as an outsider, to understand the so far unsuccessful attempts to solve important open problems.

So my question is:

• Is it possible to decribe the walls that attempts to incorporate local gauge invariance in AQFT have hit?

• What about the possibility that this is the wrong question and there is and should not be a place of local gauge invariance in AQFT?

Edit: For this question let's assume that a central goal of the reasearch program of AQFT is to be able to describe the same phenomena as the standard model of particle physics.

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retagged Mar 24, 2014
Just a comment for this moment: in a preprint by Ciolli and Ruzzi that appeared on the arxiv today (1109.4824) they make a remark that progress towards the description of massless particles in AQFT is underway. They refer to a recent talk by Doplicher, which unfortunately I have missed this talk so I cannot comment on this.

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Just to throw it in there: there are known examples of dualities, two equivalent formulations of the samd theory, one involving gauge invariance and one not. In other words, gauge invariance is a property of a theory together with a specific classical limit thereof. For any inherently non-perturbative approach gauge invariance is probably not a good guide to follow then.

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I think the open question here should be formulated -- and usually is formulated -- not as whether Yang-Mills theory "has a place" in AQFT, but whether one can abstractly characterize those local nets that arise from quantization of Yang-Mills-type lagrangians. In other words: since AQFT provides an axiomatics for QFT independently of a quantization process starting form an action functional: can one detect from the end result of quantization that it started out from a Yang-Mills-type Lagrangian?

On the other hand, we certainly expect that the quantization of any Yang-Mills-type action functional yields something that satisfies the axioms of AQFT. While for a long time there was no good suggestion for how to demonstrate this, Fredenhagen et al. have more recently been discussing how all the standard techniques of perturbative QFT serve to provide (perturbative) constructions of local nets of observables. References are collected here, see in particular the last one there on the perturbative construction of local nets of observables for QED.

In this respect, concerning Kelly's comment, one should remember that also the construction of gauge theory examples in axiomatic TQFT is not completely solved. One expects that the Reshetikhin-Turaev+construction for the modular category of $\Omega G$-representations gives the quantization of G-Chern-Simons theory, but I am not aware that this has been fully proven. And for Chern-Simons theory as an extended TQFT there has only recently be only a partial proposal for the abelian case FHLT. Finally, notice also that non-finite degrees of freedom can be incorporated here, if one passes to non-compact cobordisms (see the end of Lurie's), which in 2-dimensions are "TCFTs" that contain all the 2d TQFT models that physicists care about, such as the A-model and the B-model.

Concerning Moshe's comment: the known dualities between gauge and non-gauge theories usually involve a shift in dimension. This would still seem to allow for the question whether a net in a fixed dimension is that of a Yang-Mills-type theory.

But even if it turns out that Yang-Mills-type QFTs do not have an intrinsic characterization. their important invariant porperties should have. For instance it should be possible to tell from a local net of observables whether the theory is asymptotically free. I guess?

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answered Sep 24, 2011 by (6,025 points)
Urs, see Seiberg's duality in 4 dimensions in which both sides of the duality involve some gauge invariance, but a different one. The gauge invariance of one side is invisible on the other side - simply because all fields a already singlets. Besides, any gauge non-invariant statement is by definition unphysical, so you cannot even formulate what it means for a theory to be a "gauge theory" using only physical statements involving only observables. This suggests that gauge invariance is merely a useful tool tied directly to perturbation theory.

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Whether one can "even formulate" what it means for a theory to be a quantum Yang-Mills-type theory or a quantum Chern-Simons theory and so on is the (open) question here. It is not true that nothing about the gauge group is invariantly encoded. The invariant assigned by CS theory of course depend on this. So the fact that physical states are gauge invariant is not an argument that quantum YM does not have an intrinsic characterization.

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Since Seiberg- and Montonen-Olive and other S-dualities relate two gauge theories with each other, that does not provide an argument that quantum gauge theories don't have an intrinsic characterization. For that argument you need that one side of the duality is not a gauge theory. And of the same dimension.

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Look at the Seiberg duality examples which are simultaneously described as su(n1) theory and su(n2) theory. Any observable quantity has these two simultaneous descriptions. I think this means that you cannot resolve the difference between the two descriptions using only physically measurable input, maybe this is just lack of imagination on my part but I don't see how to get around that argument.

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But the question is: can we invariantly tell from a QFT if it is a gauge theory at all? Both the su(n1) and the su(n2) theory are, so this example would still be consistent with the answer "yes". (The answer may still be "no", but not for this reason, as far as I can see.)

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Also, in the N=4 case you see that you cannot even define the dimension of the spacetime using only physical data. In some regime of the coupling the theory is approximately local in 5 dimensions. I'd say this makes the problem I am pointing out even worse, when you move beyond perturbation theory you cannot decide for the theory on what space (and what dimensionality) it lives in.

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Well, classical gravity in AdS5 times S^5, is it a 4 dimensional non-Abelian gauge theory? turns out it is.

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As I said in reply that we are commenting on, in that class of examples the dimension is not the same. The original AQFT question only makes sense for a fixed dimension. The old problem in AQFT is: given a local net of observables on n-dimensional spacetime, did it arise as the quantization of a Yang-Mills-type theory on that spacetime? It is (another) open question to even formulate what AdS/CFT might mean in terms of the AQFT axtioms. There was once an attempt (http://ncatlab.org/nlab/show/Rehren+duality) but that didn't live up to its promise, I think.

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OK, I see your point, but personally I strongly doubt that is possible. At the very least the answer to that question should not be able to distinguish n1 from n2 in the example above. Also, I am now wondering if there are four dimensional examples of dualities between gauge and non-gauge theories, because that would pretty much settle the issue I think.

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I don't know about using local operators, but what about Wilson loops and 't Hooft operators? I don't know non-gauge theories that have this kind of operator (and they have an algebra that can at least see the center of the gauge group, which shows up in other ways too).

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That's interesting, but I am not sure if given a set of loop operators transforming under some discrete global symmetry and obeying a certain algebra, you can then unambigously identify them as Wilson and 'tHooft loop of some gauge theory, at least not without extra input. Maybe that extra input is natural in AQFT, don't know enough about the subject.

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@UrsSchreiber "...non-finite degrees of freedom can be incorporated here, if one passes to non-compact cobordisms..." The axioms of TQFT explicitly forbid non-compact cobordisms. (See Turaev's TQFT Axioms http://bit.ly/p2965P) So, what you are talking about is not an axiomatic TQFT.

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@UrsSchreiber "...the construction of gauge theory examples in axiomatic TQFT is not completely solved..." I am not sure what you mean here. Do you mean all possible gauge theory realizations of the TQFT axioms have not been constructed? If this is what you mean, then I agree, but this seems auxiliary to the main topic.

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Kelly, it is easy to have the axiomatics so that it admits non-finite degrees of freedom: simply disallow cobordisms with no outgoing (alternatively: ingoing) boundary. That's sometimes called "non-compact" TQFT. See section 4.2 of Lurie's http://www-math.mit.edu/~lurie/papers/cobordism.pdf from def. 4.2.10 on. This is certainly axiomatic TQFT. But it is more general than Turaev's definition. See also the article by Costello http://arxiv.org/PS_cache/math/pdf/0412/0412149v7.pdf that it makes the relation to.

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Kelly, the statement for (T)QFT axiomaized as cobordism representations (compact or non-compact) is similar to that of QFT axiomazized as local nets of observables: only in a small number of cases have examples of the axioms been constructed from actual quantization of an action functional. A general proof of existence or even an intrinsic characterization of those models of the axioms that arise from quantizing a gauge-theoretic Lagrangian has not been written out. This is an open problem that people are working on. See eg the FHLT reference that I provided.

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@UrsSchreiber It seems to me you are saying if you start with an Turaev axiomatic TQFT, then disallow cobordisms with no outgoing boundary, then you will arrive at a axiomatic TQFT that is more general than the Turaev axiomatic TQFT you started with. This does not make sense to me. Once you disallow certain cobordisms within Turaev's axiomatic TQFT you are by definition dealing with a less general theory.

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Kelly, no, that's not what I said. You should try the references that I gave if what I say remains unclear. But the simple idea is that there is a non-full subcategory (or sub-n-category) of the usual cobordisms category (or cobordisms n-category) that only contains the cobordisms with non-empty incoming boundary. Representations of this are TQFTs with possibly non-finite state spaces, such as, in 2d, the A-model and the B-model topological strings and more generally, every 2d QFT defined by a Calabi-Yau A-infinity algebra.

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@UrsSchreiber I admittedly haven't looked at the references you suggest as what you are claiming does not make sense to me. You say global topology changes on a manifold give rise to changes in the number of local degrees of freedom. This simply does not make any sense. Local degrees of freedom are completely independent of global topology. Global topology changes give rise to changes in the global degrees of freedom not changes in the local degrees of freedom.

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Kelly, this is simple: the fact that full TQFT state spaces are finite dimensional is because cap and cup cobordisms induce a trace on the state spaces, so traces must exist. In other words, the state space is a dualizable object. If you remove either the cap or the cup, it no longer needs to be a dualizable object, just a "Calabi-Yau object". The two references that I pointed to, by Costello and by Lurie, are foundational for the field of TQFT. If you are interested in this field, you need to take a look.

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By the way, there are no local degrees of freedom at all in TQFT.

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@UrsSchreiber Did you read my answer before you commented on it? To me it is obvious you did not. I said "...axiomatic TQFT local gauge symmetries are so 'strong' that they remove all local degrees of freedom" exactly what you are now saying after saying what I said was incorrect.

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@Urs, thanks, so a better question to ask would be how to classify nets coming from certain quantization procedures of gauge theories, or maybe to classify gauge theories which produce Haag-Kastler nets. BTW, I think it is also possible to chat in certain chat rooms here, which would be - maybe - easier for a longer conversation like the one that happened here.

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This is more of an extended comment, rather than an answer completely separate from what Urs and Moshe have already said. The axioms of AQFT are designed to capture a mathematical model of the physical observables of a theory, while OTOH gauge invariance is a feature of a formulation of a theory, though perhaps an especially convenient one. Yours and related questions are somewhat muddied by the fact that one physical theory may have several equivalent, but distinct formulations, which may also have different gauge symmetries. One example of this phenomenon is gravity, consider the metric and frame-field formulations, and another one according to Moshe is Seiberg duality. Another confounding factor is that some physical theories are only known in a formulation involving gauge symmetries (automatically rendering such formulations "especially convenient"), which naturally leads to your second question. However, one must remember that by design the gauge formulation should be visible in the AQFT framework only if it is detectable through physical observables.

Now, to be honest, I really have no idea about what is the state of the art in AQFT of figuring out when a given net of local algebras of observables admits an "especially convenient" formulation involving gauge symmetry. But I believe answering this kind of question will remain difficult until the notion of "especially convenient" is made mathematically precise. I don't know how much progress has been made on that front either. But I think a prototype of this kind of question can be analyzed, though somewhat sketchily, in the simplified case of classical electrodynamics.

Suppose we are given a local net of Poisson algebras of physical observables (the quantum counterpart would have *-algebras, but other than that, the geometry of the theory is very similar). The first step is to somehow recognize that this net of algebras is generated by polynomials in smeared fields, $\int f(F,x) g(x)$, where $g(x)$ is a test volume form, and $f(F,x)$ is some function of $F$ and its derivatives at $x$, with $F(x)$ a 2-form satisfying Maxwell's equations $dF=0$ and $d({*}F)=0$. Since we were handed the net of algebras with a given Poisson structure, as a second step we can compute the Poisson bracket $\{F(x),F(y)\}=(\cdots)$. The answer for electrodynamics would be the well known Pauli-Jordan / Lichnerowicz / causal propagator, which I will not reproduce here. Very roughly speaking, the components of $F(x)$ and the expression for the Pauli-Jordan propagator give a set of local "coordinates" on the phase space of the theory and an expression for the Poisson tensor on it. In the third step we can compute the inverse of the Poisson tensor, which if it exists would be a symplectic form. The answer for electrodynamics is well known and what's important is that the symplectic form is not given by some local expression like $\Omega(\delta F_1,\delta F_2) = \int \omega(\delta F_1, \delta F_2, x)$, where $\omega$ is a form depending only on the values of $\delta F_{1,2}(x)$ and their derivatives at $x$. Step four would consist of asking the question whether there is another choice of local "coordinates" on the phase space in which the symplectic form is local. The answer is again well known: extend the phase space by introducing the 1-form field $A(x)$ such that $F=dA$. The pullback of the symplectic form to the extended phase space now has a local expression $\Omega(\delta A_1,\delta A_2) = \int_\Sigma [{*}d(\delta A_1)(x)\wedge (\delta A_2)(x) - (1\leftrightarrow 2)]$, up to some constant factors, with $\Sigma$ some Cauchy surface. Note that $\Omega$ is no longer symplectic on the extended phase space, but only presymplectic, while its projection back to the physical phase space is. As a last step, one might try to solve the inverse problem of the calculus of variations and come up with a local action principle reproducing the the equations of motion for $A$ and the presymplectic structure $\Omega$.

Let me summarize. (1) Obtain fundamental local fields and their equations of motion. (2) Express the Poisson tensor and symplectic form in terms of local fields. (3) Introduce new fields to make the expression for the (pre)symplectic form local. (4) Obtain local action principle in the new fields. Note that gauge symmetry and all the issues associated with it appear precisely in step (3). In my limited understanding of it, the literature on AQFT has spent a significant amount of time on step (1), but perhaps not enough time on steps (2) and (3) even to formulate these problems precisely.

Finally, I should emphasize that the idea that redundant gauge degrees of freedom are introduced principally to give local structure to the (pre)symplectic structure on phase space is somewhat speculative. But it seems to fit in the field theories I am familiar with and I've not been able to identify a different yet equally competitive one.

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answered Sep 25, 2011 by (420 points)
As a comment, could one perhaps get a better view of the difficulty of "step 3" by jumping straight to a non-Abelian theory? In particular, I would have argued that the "nature" variables for the non-Abelian theories are the holonomies (intrinsically non-local), but we know that even classically one needs to smear "strongly" to get a theory. One could view this as a classical symptom of some fundamental singularity in the quantum theory (because after all the Poisson structure is the limit of the corresponding quantum geometry). Further, the electrodynamics case is then the limiting case.

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What about the possibility that this is the wrong question and there is and should not be a place of (sic) local gauge invariance in AQFT?

I'd guess this hinges upon how one views AQFT. One can view AQFT in one of two ways:

• AQFT as a theory needs to correspond to nature.
• AQFT as a theory need not correspond to nature.

If AQFT needs to correspond to nature, then it should incorporate local gauge invariance, as nature incorporates local gauge invariance. (Note, "incorporate" here could mean include a mechanism which at "low energies" looks like local gauge invariance.)

If AQFT need not correspond to nature, then it need not incorporate local gauge invariance.

With that in mind I would also add that axiomatic TQFT includes local gauge symmetries without problems. In fact, axiomatic TQFT local gauge symmetries are so "strong" that they remove all local degrees of freedom.

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answered Sep 23, 2011 by (220 points)
Ok, maybe I should explain my own viewpoint of what AQFT is: That it should be able to describe the same phenomena as the standard model.

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@TimvanBeek I guess your question then is almost equivalent to: "Do we need local gauge symmetry?" I guess the obvious answer is: "No." We have some theory with a local gauge invariance, we fix the local gauge invariance, and work in that particular gauge. Ugly, unilluminating, but it would work. I guess my real point is that your second question is rather "ill-defined" and maybe needs to be sharpened.

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Well, maybe, but how? But please note that the question is not "do we need local gauge symmetry" but "do we need local gauge symmetry in the framework of AQFT", with the latter being somewhat more precise than the former.

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